a. Suppose that
S_{2} X_{1}
+ X_{2}, where
X_{1} and
X_{2}
are independent Cauchy random variables, each having pdf
*f*_{xi}(.).
Using the integral identity

show that
S_{2} has a pdf 2/[(4 + y^{2})].

b. Proceeding by induction, show that

**S**_{n}
X_{1}
+ X_{2} + . . . +
X_{n}. (all
X_{i} independent)

has a pdf **n / [(n**^{2}** + ***Y*^{2}**)]**.

c. Thus, verify that the average of *n*
independent Cauchy samples
(i.e., V_{n}**
**S_{n} / n) has a
Cauchy pdf 1 **/ [(n**^{2}** + ***Y*^{2}
**)]**. Thus,
"averaging together" a number of independent Cauchy
samples yields a pdf for the
average identical to that of any one of the individual samples.
(This result contrasts
sharply to most random variables, for which averaging of n
independent samples reduces the
variance by a factor of n^-1.)